Comprehensive Graphical Testing for Exponentially Increasing Data

JSM 2023, Toronto, CA

Emily Robinson

Cal Poly - San Luis Obispo

Reka Howard

University of Nebraska - Lincoln

Susan VanderPlas

University of Nebraska - Lincoln

Outline

Motivation and Background
Comprehensive Graphical Testing
Research Goals
- Perception through Lineups
- Prediction through ‘You Draw It’
- Numerical Translation and Estimation
Overall Conclusions and Discussion

Exponential Growth

Von Bergmann (2021)

Benefits and pitfalls of log scales

Burn-Murdoch et al. (2020)

The motivation for this work came from the large impact graphics and charts have had during the COVID-19 pandemic. Many of these graphics helped guide decision makers to implement policies such as shut-downs or mandated mask wearing, as well as facilitated communication with the public to increase compliance.

As graphics began to play an important role in sharing information with the public, it was crucial to start asking what made some of these charts better than others? Creators of the graphics were faced with many design choices in order to ensure their charts were effective at accurately communicating the current status of the pandemic.

In order to make educated decisions when designing a chart, we need to establish guidelines through experimentation in order to ensure the graphic is effective at communicating the intended results.

A major issue we encountered in the creation of COVID-19 plots was how to display data from a wide range of values.

Problem: Data which spans several orders of magnitude shown on its original scale compresses the smaller magnitudes into relatively little area.
Solution: Use of a log scale transformation; alters the contextual appearance of the data.

Benefits were seen in spring 2020, during the early stages of the COVID-19 pandemic.

Large magnitude discrepancies in case counts at a given time point between different geographic regions.
Log scale transformations were usefulness for showing case count curves for areas with few cases and areas with many cases within one chart.

Pitfalls were exposed as the pandemic evolved, and the case counts were no longer spreading exponentially.

Graphs with linear scales seemed more effective at spotting early increases in case counts that signaled more localized outbreaks.
The effect of the linear scale appears to evoke a stronger reaction from the public than the log scale.

While this is one such example, there is a long history of using log scales to display results in ecology, psychophysics, engineering, and physics. Given the widespread use of logarithmic scales, it is important to understand the implications of their use in order to provide guidelines for best use.

Testing statistical graphics

Evaluate design choices and understand cognitive biases through the use of visual tests.

Could ask participants to:

identify differences in graphs.
read information off of a chart accurately.
use data to make correct real-world decisions.
predict the next few observations.

All of these types of tests require different levels of use and manipulation of the information presented in the chart.

Task complexity

Carpenter and Shah (1988) identifies pattern recognition, interpretative processes, and integrative processes as strategies and processes required to complete tasks of varying degrees of complexity.

Pattern recognition requires the viewer to encode graphic patterns.
Interpretive processes operate on those patterns to construct meaning.
Integrative processes then relate the meanings to the contextual scenario as inferred from labels and titles.

Research objectives

Big Idea: Are there benefits to displaying exponentially increasing data on a log scale rather than a linear scale?

Perception through Lineups – tests an individual’s ability to perceptually differentiate exponentially increasing data with differing rates of change on both the linear and log scale.
Prediction with ‘You Draw It’ – tests an individual’s ability to make predictions for exponentially increasing data.
Estimation by Numerical Translation – tests an individual’s ability to translate a graph of exponentially increasing data into real value quantities.

The series of graphical tests were conducted through an RShiny application found at https://emily-robinson.shinyapps.io/perception-of-statistical-graphics-log/.

In this research, we conducted a series of three graphical studies to evaluate the impact displaying data on the log scale has on human perception of exponentially increasing trends compared to displaying data on the linear scale. Each study was related to a different graphical task, each requiring a different level of interaction and cognitive use of the data being presented.

The first experiment evaluated whether our ability to perceptually notice differences in exponentially increasing trends is impacted by the choice of scale. These perceptual differences are identified through pattern recognition.
The second study tested an individuals ability to make predictions for exponentially increasing data and required integrative processes by operate on the patterns to construct meaning.
The last and third study tested an individual’s ability to translate a graph of exponentially increasing data into real value quantitites and extend these estmiates by making comparisons.

Combined, the three studies provide a comprehensive evaluation of the impact of displaying exponentially increasing data on a log scale as it relates to perception, prediction, and estimation. The results of these studies help us make recommendations and provide guidelines for the use of log scales.

Perception through lineups

Lineup experimental task

Study Participant Prompt: Which plot is most different?

Lineup study design

Treatment Design: Target Panel gets model A and Null Panels get model B

\(3!\cdot 2!= 6\) curvature combinations

\(\times 2\) lineup data sets per combination \(=\) 12 test data sets

\(\times 2\) scales (log & linear) \(=\) 24 different lineup plots

Experimental Design: 12 lineup plots per participant

\(6\) test parameter combinations per participant \(\times 2\) scales \(= 12\) test lineups

We used a ‘goldilocks’ type of approach to select coefficients related to three levels of curvature (low, medium, and high) for our data simulation.

The lineups were then created by looking at combinations of each of these curvature levels where the data in the target panel was simulated with equation A and the data in the null panels were simulated with equation B for a total of 6 curvature combinations.

We simulated 2 replicate data sets of each curvature combinations and each data set was plotted on botht he log and linear scales for a total of 24 different lineup plots to select from. Each participant ended up evaluating 12 lineup plots and ideally saw one of each of the 6 curvature combinations on both scales.

Lineup results

Prediction through ‘You Draw It’

‘You Draw It’ experimental task

Study Participant Prompt: Use your mouse to fill in the trend in the yellow box region.

In 2015, the New York Times introduced an interactive feature, called ‘You Draw It’, where readers input their own assumptions about various metrics and compare how these assumptions relate to reality. The Times team utilizes Data Driven Documents (D3) that allows readers to predict these metrics through the use of drawing a line on their computer screen with their mouse. We adapted this feature as a way to measure the patterns we see in data and compare intuitive visual model fitting and statistical model fitting. In the study, participants were shown a series of ‘You Draw It’ interactive task plots and asked to “Use your mouse to fill in the trend in the yellow box region.” The yellow box region moves along as the participant draws their trend-line until the yellow region disappears, providing a visual cue for the task.

‘You Draw It’ study design

Low Growth Rate

50% Truncation

Low Growth Rate

75% Truncation

High Growth Rate

50% Truncation

High Growth Rate

75% Truncation

‘You Draw It’ prediction results

‘You Draw It’ prediction results

Numerical Translation and Estimation

Integrative processes

Graph Comprehension

The three behaviors related to graph comprehension involve (Curcio 1987; Friel, Curcio, and Bright 2001; Glazer 2011; Jolliffe 1991; Wood 1968)

literal reading of the data (elementary level)
reading between the data (intermediate level)
reading beyond the data (advanced level).

Estimation Biases

Anchoring Tan and Benbasat (1990) and rounding to multiples of five or ten Myers (1954) arise in open-ended estimation tasks.
Scale and axis labels are other critical factors in estimation accuracy (Dunham and Osborne 1991; Beeby and Taylor 1973“; Leinhardt, Zaslavsky, and Stein 1990).

This study involved integrative processes which required viewers to translate the visual features into conceptual relations by interpreting titles, labels, and scales.

An important consideration in understanding graph comprehension is the questions being asked of the viewer. Three levels of graph comprehension have emerged from mathematics education research: literal reading of the data (elementary level), reading between the data (intermediate level), and reading beyond the data (advanced level).

While not exclusive to extracting numerical values from charts, mathematics education also places an emphasis on quantitative estimation skills such as providing an estimated length of a string or weight of a box.

With open ended estimation tasks, certain biases including anchoring and rounding to multiples of five or ten arise.

Beeby & Taylor (1973) found that when asked to read data from line graphs, viewers consistently misread the y-axis scale; when alternate grid lines were labeled, the unlabeled grid lines were read as halves.
The choice of scale can change the shape of a graph, thus creating a conceptual demand for the viewer when constructing a mental image of the graph (Leinhardt, Zaslavsky, & Stein, 1990)

Simulated Data

The time unit labels on the x-axis reflected 0 to 50 ABY (After Battle of Yavin) for the Ewok scenario and were adjusted to 4500 to 4550 stardates for the Tribble Scenario to align with the associated popular media depiction of each figure. s Two unique data sets were generated with the same underlying parameter coefficients, but different errors randomly generated from the same error distribution. Notice in particular that some of the simulated points fall farther from the “true value” or underlying curve. For example, year 40 in data set 2.

This figure displays scatter plots of the two unique data sets on both the linear and log base two scales. A log base two was selected in order to aid in participants estimation of time until the population doubled in the third intermediate question.

I will refer to the time unit as year, but during the study, the time unit labels on the x-axis aligned with the associated popular media depiction of each figure. This was also meant to disguise the use of the same underlying data simulation model and estimation questions across both scenarios.

Questioning

Question type	Tribble scenario	Ewok scenario
Open Ended	Between stardates 4530 and 4540, how does the population of Tribbles change?	Between 30 and 40 ABY, how does the population of Ewoks change?
Elementary Q1	What is the population of Tribbles in stardate 4510?	What is the population of Ewoks in 10 ABY?
Elementary Q2	In what stardate does the population of Tribbles reach 4,000?	In what ABY does the population of Ewoks reach 4,000?
Intermediate Q1	From 4520 to 4540, the population increases by ____ Tribbles.	From 20 ABY to 40 ABY, the population increases by ____ Ewoks.
Intermediate Q2	How many times more Tribbles are there in 4540 than in 4520?	How many times more Ewoks are there in 40 ABY than in 20 ABY?
Intermediate Q3	How long does it take for the population of Tribbles in stardate 4510 to double?	How long does it take for the population of Ewoks in 10 ABY to double?

We selected two scenarios and provided text to participants describing a context in which populations of fuzzy fictional characters from star trek and star wars are growing exponentially over periods of time. We wanted to go with a fun context to avoid any personal opinions or biases affecting the results.

We then selected 6 questions related to each scenario. The open ended question was meant to prompt the participants to take time to inspect the graph while the two intermediate questions are related to the literal reading and estimation of population and year respectively. There were three intermediate questions selected to address the additive increase in population between two years, the multiplicative change in population between two years, and the amount of time it took the population to double from a certain year.

We did not focus on advanced level questioning since extrapolation and interpolation were addressed in the second study.

Each of these questions will again be presented in the following slides of results.

During the study, participants were first shown two examples with associated sketches that were available to them throughout the study.

The scale of the graphic and data set displayed was randomly assigned to scenarios for each individual. + For instance, a participant may have seen a scatter plot of data set two displayed on the linear scale paired with the Ewok scenario text and a scatter plot of data set one displayed on the log scale paired with the Tribble scenario text.

The order of the two scenarios and their assigned data set and scale was randomly assigned to each individual.

In each scenario, participants were first asked an open ended question, which required them to spend time exploring the data displayed in the graphic, followed by a random order of two elementary level questions and three intermediate level questions.

Estimation strategy

Estimation results

Open ended

“Between year 30 and 40, how does the population of [creatures] change?”

Estimation of population (Elementary Q1)

“What is the population in year 10?”

In order to examine the effect of scale on literal reading of the data, participants were asked, “What is the population in year 10?”

Here we see a sketch of the estimation procedure. Participants first locate 10 along the x-axis and move upward until they believe they have found the correct location on the curve; then participants look to the y-axis for their estimated population.

We used density plots were to illustrate the distribution of the quantitative estimates for the population in year 10 provided by participants. The colors are associated to the scale - linear (blue) and log (orange) - and vertical lines indicate the true value based on the underlying model equation (black solid), closest point value based on the simulated data set (black dashed), and grid lines shown on the graphs (blue/orange dotted). A jittered rug plot along the x-axis shows where participant estimates were made. The two unique data sets are shown separately.

The figure reveals a larger variance in quantitative population estimates made on the linear scale compared to the log scale. As expected, it is clear that participants were anchoring to grid lines and base ten values as highlighted by the high density of estimates at 512 and 500 on the log scale as well as local maximums near multiples of ten such as 500 and 1,000.

One such explanation for this difference might be the location of year 10 along the x-axis. This year corresponds to a low magnitude where the population is condensed in a small region on the linear scale as opposed to later in time when larger magnitudes in population can be seen. Thus, it is important to evaluate the accuracy of estimates along the full domain.

Estimation of population (First level estimates)

In two estimation questions related to intermediate level reading between the data, participants are asked to provide an increase and change in population between years 20 and 40, thus requiring participants to make first-level estimates at these locations. To understand the effect of the location along the domain and in turn the magnitude of the population being estimated, we extracted first-level estimates for years 20 and 40 from participant calculations and scratch work.

These figures show evaluations of participants estimates of the population at years 10, 20, and 40 for each data set separately. Spaghetti plots are displayed on the linear scale (top) and log scale (bottom) with both scale estimates shown on each - linear (blue), log (orange). The year was calculated from the underlying model equation based on the population estimate provided by the participant. Gray arrows indicate the true value and closest point value as demonstrated in the sketch.

As the year increases, we observe an increased accuracy in estimates made on the linear scale while estimates made on the log scale suffer in accuracy due to strong anchoring to grid lines and the larger quantitative difference between grid lines as population magnitudes increase. + For instance, on the log scale, there was a tendency to overestimate the population for year 20 from data set one, underestimate the population for year 20 from data set two, and overestimate the population for year 40 from data set two. This provides argument that estimates are highly subjective to the particular data set.

This is important to recognize becuase inaccurate first-level estimations can lead to consequences in estimations which require participants to make comparisons between two points (e.g. Intermediate Q1 and Q2).

Estimation of population (Scratchwork)

Participant calculations and scratch work provides support that participants equated halfway spatially as halfway numerically.

2048 − 1024 = 1024

1024/2 = 512

512 + 1024 = 1536

32768 − 16384 = 16384

16384 ∗ 2 = 32768

16384/2 = 8192

8192 + 16384 = 24576.

Estimation of time (Elementary Q2)

“In what year does the population reach 4,000?”

Additive increase in population (Intermediate Q1)

From 20 to 40, the population increases by ____ [creatures].

Intermediate level questions required participants to read between the data and make comparisons between points. Participants were asked, “From 20 to 40, the population increases by _______ [creatures].” (Fig. 4.10). The questioning was selected carefully to prompt participants to make an additive comparison of populations between two years.

We saw during the sketch on the previous slide for first level estimates that this procedure required participants to make first-level population estimates at years 20 and 40, then calculate the difference between the two values by subtraction.

These two figures display the density for estimated increases in population as made by participants for data set one and two respectively. + There were a considerable amount of estimated increases near zero indicating that some participants were misinterpreting the value they were asked to estimate. + Estimates for the increase in population between year 20 and year 40 was distinctly more accurate for estimates made on the linear scale as indicated by the peak density occurring near the closest point and true value vertical lines. + The slight shifts in the density on the log scale suggest participants are making inaccurate first-level estimates. One explanation might be that participants were anchoring to the grid lines much stronger on the log scale as opposed to being more likely to adjust their estimates between grid lines on the linear scale.

Here we also see the subjectivity of results due to the unique data points as the point associated with year 40 appears much further from the underlying data trend for data set 2 than it does for data set 1.

Multiplicative change in population (Intermediate Q2)

How many times more [creatures] are there in 40 than in 20?

Participants were asked, “How many times more [creatures] are there in 40 than in 20?” The questioning was selected carefully to prompt participants to make a multiplicative comparison between two years.

Here we see a similar sketch as before where participants make first-level population estimates at years 20 and 40, but then calculate the ratio between the two values rather than the difference.

The density plots shown for each data set separately, show participants tended to be make more accurate and less variable estimates on the log scale than on the linear scale. Evaluating some of the scratch-work for this question, it was interesting to see how many participants used a “guess-and-check” sort of method of multiplying their population estimate from year 20 by different multipliers until they felt the result matched the population at year 40.

Misunderstanding of log logic

Misinterpretation and miscalculations in Intermediate Q1 and Q2

Additive Increase Incorrect Logic

2048 − 1024 = 1024

1024 + 512 = 1536

16384 − 1536 = 14848

14848/1536 = 9.67.

Multiplicative Change Incorrect Logic

16384 − 1536 = 14848

Time until population doubles (Intermediate Q3)

How long does it take for the population in year 10 to double?

An alternative multiplicative comparison between two points was to determine the amount of time it took for a value to double. Participants were asked, “How long does it take for the population in year 10 to double?”

This required participants to make a first-level population estimate at year 10, then double this population and estimate the year in which that occurs. Lastly, participants subtracted year 10 to determine how long it took for the population to double. Another estimation strategy could involve participants spatially judging the double of population on the log base2 scale; however, examining the scratch-work revealed most participants took the first approach to estimation.

The estimated number of years until the population doubled from a reference year of 10 resulted in lower variability on the log scale as opposed to a larger variability on the linear scale. One explanation might be the low magnitude of population in year 10 and further exploration would be needed to justify for other reference years. Common responses strongly suggest participants were anchoring to multiples of 5 years.

Main take-aways

Understanding log logic is difficult.
Accuracy greatly depends on the location of the value being estimated in relation to the magnitude.
Strong anchoring and rounding effects.
- Participants were resistance to estimate between grid lines on the log scale.
- Inaccurate representation of equating spatial distance to quantitative difference.
Inaccurate first level estimations can lead to consequences in estimations which require participants to make comparisons between two points.
Estimates were subjective to the simulated data set.

This study was intended to aid in understanding the cognitive implications of displaying exponentially increasing data on a log scale.

Some of the main take aways from this study are shown here. Overall, we provide argument that understanding log logic is difficult and that the accuracy greatly depends on the location of the value being estimated in relation to the magnitude.

We found biases related to strong anchoring and rounding effects as participants were resistant to estimate between grid lines on the log scale and had inaccurate representations of equating that spatial distance to the quantitative difference.

This inaccuracy in first-level estimations showed a clear consequence in estimations which required participants to make comparisons between two points.

The use of two unique data sets allowed us to look at how estimates are very subjective to the simulated data set as some participants are actually reading the data off of the plot and not basing estimates off the underlying trend.

Conclusion and Discussion

Conclusions

1. Perception through Lineups

Perceptual differences result from the contextual appearance (depends on choice of scale) of the trends.

2. Prediction through ‘You Draw It’

Clear underestimation of forecasting trends with high exponential growth rates when participants were asked to make predictions on the linear scale.

3. Numerical Translation and Estimation

Log logic is difficult and that we often misinterpret and miscalculate multiplicative reasoning.
Estimation accuracy for small magnitudes was improved by the use of the log scale, but sacrifices in accuracy on the log scale became apparent as magnitudes increased leading to advantages on the linear scale.

This research evaluated the use of log scales to display exponentially increasing data from three different angles and levels of complexity: perception, prediction, and estimation. Each study provided us insight into the advantages and disadvantages of displaying exponentially increasing data on a log scale and in what context each choice of scale might be more appropriate.

The first study laid the foundation for the future studies. Results from this study indicated the choice of scale changes the contextual appearance of the data leading to slight perceptual advantages for both scales depending on the curvatures of the trend lines being compared.

The second study tested participant’s abilities to make forecast predictions for exponentially increasing trends on both scales. The results from the analysis showed a clear underestimation of forecasting trends with high exponential growth rates when participants were asked to make predictions on the linear scale and improvements in forecasts were made when participants were asked to make predictions on the log scale.

The third study evaluated graph comprehension as it relates to the contextual scenario of the data shown. Overall, our results suggested that log logic is difficult and that anchoring and rounding biases result in a sacrifice in accuracy in estimates made on the log scale for large magnitudes.

Overall Recommendations

Perceptual advantages of the use of log scales due to the change in contextual appearance.
Our understanding of log logic is flawed when translating the information into context.
We recommend consideration of both user needs and graph specific tasks when presenting data on the log scale.
Caution should be taken when interpretation of large magnitudes is required, but advantages may appear when it is necessary to visually identify and interpret small magnitudes on the chart.

The results are instrumental in establishing guidelines for making design choices about scale which result in data visualizations effective at communicating the intended results.

We recommend understanding the contextual appearance log scales have on the data being displayed and the perceptual advantages and disadvantages that come with each. It is also important to recognize that our understanding of log logic is flawed when translating this information into context.

In general, it is important to consideration of both user needs and graph specific tasks when presenting data on the log scale. Caution should be taken when interpretation of large magnitudes is required, but advantages may appear when it is necessary to visually identify and interpret small magnitudes on the chart.

Future work

Follow-up studies for this research could provide further insight into the strategies of estimation by providing users the ability to interact with the graph with visual aids such as the arrows shown in sketches from the estimation study and how we make decisions based on data displayed on a log scale to evaluate the effect of scale on risk adversity.

This research stands as a model for conducting an extensive series of graphical tasks on the same type of data and plot in order to gain a comprehensive understanding of both the perceptual and cognitive implications of the design choices.

Combine participant data across all three studies to conduct a comprhensive analysis.
Potential follow-up studies:
- Providing users the ability to interact with the graph with visual aids in order to better understand strategies of estimation.
- Additional study which focuses on how we make decisions based on data displayed on a log scale in order to evaluate the effect of scale on risk adversity.
Expand the use of log scales to data which does not follow an exponential trend, but where log scales are otherwise appropriate.
Evaluate the implications of transforming or displaying the \(x\)-axis on a log scale.
More widespread use of comprehensive graphical studies conducted on the same type of data and plot.

References

Beeby, AW, and HPJ Taylor. 1973. “How Well Can We Use Graphs.” The Communicator of Scientific and Technical Information 17: 7–11.

Burn-Murdoch, John, Caroline Nevitt, Cale Tilford, Andrew Rininsland, Joanna Kao, Oliver Elliott, Emma Lewis, Brooke Fox, and Martin Stabe. 2020. “Coronavirus Tracked: Has the Epidemic Peaked Near You?” Coronavirus Chart: See How Your Country Compares. Financial Times. https://ig.ft.com/coronavirus-chart/?areas=eur.

Carpenter, Patricia A, and Priti Shah. 1988. “A Model of the Perceptual and Conceptual Processes in Graph Comprehension,” 26.

Curcio, Frances R. 1987. “Comprehension of Mathematical Relationships Expressed in Graphs.” Journal for Research in Mathematics Education 18 (5): 382–93. https://doi.org/10.2307/749086.

Dunham, Penelope, and Alan Osborne. 1991. “Learning How to See: Students Graphing Difficulties.” Focus on Learning Problems in Mathematics 13 (4): 35–49.

Friel, Susan N., Frances R. Curcio, and George W. Bright. 2001. “Making Sense of Graphs: Critical Factors Influencing Comprehension and Instructional Implications.” Journal for Research in Mathematics Education 32 (2): 124–58. https://doi.org/10.2307/749671.

Glazer, Nirit. 2011. “Challenges with Graph Interpretation: A Review of the Literature.” Studies in Science Education 47 (2): 183–210. https://doi.org/10.1080/03057267.2011.605307.

Jolliffe, Flavia. 1991. “Assessment of the Understanding of Statistical Concepts.” In Proceedings of the Third International Conference on Teaching Statistics, 1:461–66.

Leinhardt, Gaea, Orit Zaslavsky, and Mary Stein. 1990. “Functions, Graphs, and Graphing: Tasks, Learning, and Teaching.” Review of Educational Research 60 (1): 1–64.

Myers, Robert. 1954. “Accuracy of Age Reporting in the 1950 United States Census.” Journal of the American Statistical Association 49 (268): 826–31.

Tan, Joseph, and Izak Benbasat. 1990. “Processing of Graphical Information: A Decomposition Taxonomy to Match Data Extraction Tasks and Graphical Representations.” Information Systems Research, 416–39.

Von Bergmann, Jens. 2021. “Xkcd_exponential: Public Health Vs Scientists.” mountainMath. GitHub. March 17, 2021. https://github.com/mountainMath/xkcd_exponential.

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Appendix

Lineup GLMM

Define \(Y_{ijkl}\) to be the event that participant \(l\) correctly identifies the target plot for data set \(k\) with curvature \(j\) plotted on scale \(i\).

\[\text{logit }P(Y_{ijk}) = \eta + \delta_i + \gamma_j + \delta \gamma_{ij} + s_l + d_k\]

where

\(\eta\) is the baseline average probability of selecting the target plot.
\(\delta_i\) is the effect of the log/linear scale.
\(\gamma_j\) is the effect of the curvature combination.
\(\delta\gamma_{ij}\)is the two-way interaction effect of the scale and curvature.
\(s_l \sim N(0,\sigma^2_\text{participant})\), random effect for participant characteristics.
\(d_k \sim N(0,\sigma^2_{\text{data}})\), random effect for data specific characteristics.

We assume that random effects for data set and participant are independent.

Feedback data

For each participant, the final data set used for analysis contains:

\(x_{ijklm}\), \(y_{ijklm,drawn}\), and \(\hat y_{ijklm,NLS}\)

for:

growth rate \(i = 1,2\),
point truncation \(j = 1,2\),
scale \(k = 1,2\),
participant \(l = 1,...N_{participant}\), and
\(x_{ijklm}\) value \(m = 1, ...,4 x_{max} + 1\).

Vertical residuals between the drawn and fitted values were calculated as: + \(e_{ijklm,NLS} = y_{ijklm,drawn} - \hat y_{ijklm,NLS}\).

Prediction GAMM

Fit separate for each growth rate (low = 1, high = 2), the GAMM equations for residuals are given by:

\[\begin{equation} e_{1jklm,NLS} = \tau_{1jk} + s_{1jk}(x_{1jklm}) + p_{l} + s_{l}(x_{1jklm}) \end{equation}\] \[\begin{equation} e_{2jklm,NLS} = \tau_{2jk} + s_{2jk}(x_{2jklm}) + p_{l} + s_{l}(x_{2jklm}) \end{equation}\]

where

\(e_{\cdot jklm,NLS}\) is the residual between the drawn \(y\)-value and fitted \(y\)-value for the \(l^{th}\) participant, \(m^{th}\) increment, and \(\cdot jk^{th}\) treatment combination
\(\tau_{\cdot jk}\) is the intercept for the \(j^{th}\) point truncation, and \(k^{th}\) scale treatment combination
\(s_{\cdot jk}\) is the smoothing spline for the \(\cdot jk^{th}\) treatment combination
\(x_{\cdot jklm}\) is the \(x\)-value for the \(l^{th}\) participant, \(m^{th}\) increment, and \(\cdot jk^{th}\) treatment combination
\(p_{l} \sim N(0, \sigma^2_\text{participant})\) is the error due to the \(l^{th}\) participant’s characteristics
\(s_{l}\) is the random smoothing spline for the \(l^{th}\) participant.